canonical form of element of number field

Theorem.  Let $\vartheta$ be an algebraic number of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) $n$.  Any element $\alpha$ of the algebraic number field $\mathbb{Q}(\vartheta )$ may be uniquely expressed in the canonical form

$\alpha =c_0+c_1\vartheta +c_2{\vartheta }^2+\mathrm{\dots }+c_{n-1}{\vartheta }^{n-1}$ (1)

where the numbers $c_i$ are rational.

Proof.  We start from the fact that $\mathbb{Q}(\vartheta )$ consists of all expressions formed of $\vartheta$ and rational numbers using arithmetic operations (no divisor (http://planetmath.org/Division) must vanish); such expressions lead always to the form

$\alpha ={\displaystyle \frac{a(\vartheta )}{b(\vartheta )}}$ (2)

where the numerator and the denominator are polynomials in $\vartheta$ with rational coefficients (which can, in fact, be chosen integers).

So, let $\alpha$ in (2) an arbitrary element of the field $\mathbb{Q}(\vartheta )$.  Denote by $f(x)$ the minimal polynomial of $\vartheta$ over $\mathbb{Q}$.  Since  $b(\vartheta )\ne 0$,  the polynomial $f(x)$ does not divide (http://planetmath.org/DivisibilityInRings) $b(x)$, and since $f(x)$ is irreducible (http://planetmath.org/IrreduciblePolynomial2), the greatest common divisor (http://planetmath.org/PolynomialRingOverFieldIsEuclideanDomain) of $f(x)$ and $b(x)$ is a constant polynomial, which can be normed to 1.  Thus there exist the polynomials $\phi (x)$ and $\psi (x)$ of the ring $\mathbb{Q}[x]$ such that

$$\phi (x)f(x)+\psi (x)b(x)\equiv \mathrm{\hspace{0.33em}1}.$$

Especially

$$\phi (\vartheta )\underset{=\mathrm{\hspace{0.17em}0}}{\underbrace{f(\vartheta )}}+\psi (\vartheta )b(\vartheta )=\mathrm{\hspace{0.33em}1},$$

whence

$$\frac{1}{b(\vartheta )}=\psi (\vartheta )$$

and consequently

$$\alpha =\frac{a(\vartheta )}{b(\vartheta )}=a(\vartheta )\psi (\vartheta ):={\psi }_1(\vartheta ).$$

Hence, $\alpha$ is a polynomial in $\vartheta$ with rational coefficients.

Let now

Denote

$$r(x):=c_0+c_{1}x+\mathrm{\dots }+c_{n-1}{x}^{n-1}\in \mathbb{Q}[x].$$

It follows that

$$\alpha =r(\vartheta )=c_0+c_1\vartheta +\mathrm{\dots }+c_{n-1}{\vartheta }^{n-1},$$

whence (1) is true.

Suppose that we had also

$$\alpha =s(\vartheta )=d_0+d_1\vartheta +\mathrm{\dots }+d_{n-1}{\vartheta }^{n-1}$$

with every $d_i$ rational.  This implies that

$$(c_{n-1}-d_{n-1}){\vartheta }^{n-1}+\mathrm{\dots }+(c_1-d_1)\vartheta +(c_0-d_0)=\mathrm{\hspace{0.33em}0},$$

i.e. that $\vartheta$ satisfies the equation

$$(c_{n-1}-d_{n-1})x^{n-1}+\mathrm{\dots }+(c_1-d_1)x+(c_0-d_0)=\mathrm{\hspace{0.33em}0}$$

with rational coefficients and degree less than $n$.  Because the degree of $\vartheta$ is $n$, it is possible only if all differences $c_i-d_i$ vanish.  Thus

$$d_0=c_0,d_1=c_1,\mathrm{\dots },d_{n-1}=c_{n-1},$$

i.e. the (1) is unique.

Note 1.  The polynomial $c_0+c_{1}x+\mathrm{\dots }+c_{n-1}{x}^{n-1}$ is called the canonical polynomial of the algebraic number $\alpha$ with respect to the primitive element (http://planetmath.org/SimpleFieldExtension) $\vartheta$.

Note 2.  The theorem allows to denote the field $\mathbb{Q}(\vartheta )$ similarly as polynomial rings: $\mathbb{Q}[\vartheta ]$.

Note 3.  When allowed, unlike in (1), higher powers of the primitive element $\vartheta$ (whose minimal polynomial is $x^n+a_1{x}^{n-1}+\mathrm{\dots }+a_n$), one may unlimitedly write different sum of $\alpha$, e.g.

$\alpha$ $\mathrm{\hspace{0.33em}}=(c_0+c_1\vartheta +\mathrm{\dots }+c_{n-1}{\vartheta }^{n-1})+({\vartheta }^n+a_1{\vartheta }^{n-1}+\mathrm{\dots }+a_n)$

$\mathrm{\hspace{0.33em}}=(c_0+a_n)+\mathrm{\dots }+(c_{n-1}+a_1){\vartheta }^{n-1}+{\vartheta }^n.$

Title	canonical form of element of number field
Canonical name	CanonicalFormOfElementOfNumberField
Date of creation	2013-03-22 19:08:00
Last modified on	2013-03-22 19:08:00
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	17
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11R04
Related topic	Canonical
Related topic	SimpleFieldExtension
Related topic	CanonicalBasis
Related topic	IntegralBasis
Defines	canonical form
Defines	canonical form in number field
Defines	canonical polynomial